# Binary trees

A <u>binary tree</u> is a nonlinear data structure representing relationships between ancestors and descendants. It embodies the divide-and-conquer idea of splitting in two. Like a linked list, it is built from nodes, each containing a value and references to its left and right children.

=== "Python"

    ```python title=""
    class TreeNode:
        """Binary tree node class"""
        def __init__(self, val: int):
            self.val: int = val                # Node value
            self.left: TreeNode | None = None  # Left child reference
            self.right: TreeNode | None = None # Right child reference
    ```

=== "C++"

    ```cpp title=""
    /* Binary tree node struct */
    struct TreeNode {
        int val;          // Node value
        TreeNode *left;   // Left child pointer
        TreeNode *right;  // Right child pointer
        TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
    };
    ```

=== "Java"

    ```java title=""
    /* Binary tree node class */
    class TreeNode {
        int val;         // Node value
        TreeNode left;   // Left child reference
        TreeNode right;  // Right child reference
        TreeNode(int x) { val = x; }
    }
    ```

=== "C#"

    ```csharp title=""
    /* Binary tree node class */
    class TreeNode(int? x) {
        public int? val = x;    // Node value
        public TreeNode? left;  // Left child reference
        public TreeNode? right; // Right child reference
    }
    ```

=== "Go"

    ```go title=""
    /* Binary tree node struct */
    type TreeNode struct {
        Val   int
        Left  *TreeNode
        Right *TreeNode
    }
    /* Constructor */
    func NewTreeNode(v int) *TreeNode {
        return &TreeNode{
            Left:  nil, // Left child pointer
            Right: nil, // Right child pointer
            Val:   v,   // Node value
        }
    }
    ```

=== "Swift"

    ```swift title=""
    /* Binary tree node class */
    class TreeNode {
        var val: Int // Node value
        var left: TreeNode? // Left child reference
        var right: TreeNode? // Right child reference

        init(x: Int) {
            val = x
        }
    }
    ```

=== "JS"

    ```javascript title=""
    /* Binary tree node class */
    class TreeNode {
        val; // Node value
        left; // Left child pointer
        right; // Right child pointer
        constructor(val, left, right) {
            this.val = val === undefined ? 0 : val;
            this.left = left === undefined ? null : left;
            this.right = right === undefined ? null : right;
        }
    }
    ```

=== "TS"

    ```typescript title=""
    /* Binary tree node class */
    class TreeNode {
        val: number;
        left: TreeNode | null;
        right: TreeNode | null;

        constructor(val?: number, left?: TreeNode | null, right?: TreeNode | null) {
            this.val = val === undefined ? 0 : val; // Node value
            this.left = left === undefined ? null : left; // Left child reference
            this.right = right === undefined ? null : right; // Right child reference
        }
    }
    ```

=== "Dart"

    ```dart title=""
    /* Binary tree node class */
    class TreeNode {
      int val;         // Node value
      TreeNode? left;  // Left child reference
      TreeNode? right; // Right child reference
      TreeNode(this.val, [this.left, this.right]);
    }
    ```

=== "Rust"

    ```rust title=""
    use std::rc::Rc;
    use std::cell::RefCell;

    /* Binary tree node struct */
    struct TreeNode {
        val: i32,                               // Node value
        left: Option<Rc<RefCell<TreeNode>>>,    // Left child reference
        right: Option<Rc<RefCell<TreeNode>>>,   // Right child reference
    }

    impl TreeNode {
        /* Constructor */
        fn new(val: i32) -> Rc<RefCell<Self>> {
            Rc::new(RefCell::new(Self {
                val,
                left: None,
                right: None
            }))
        }
    }
    ```

=== "C"

    ```c title=""
    /* Binary tree node struct */
    typedef struct TreeNode {
        int val;                // Node value
        int height;             // Node height
        struct TreeNode *left;  // Left child pointer
        struct TreeNode *right; // Right child pointer
    } TreeNode;

    /* Constructor */
    TreeNode *newTreeNode(int val) {
        TreeNode *node;

        node = (TreeNode *)malloc(sizeof(TreeNode));
        node->val = val;
        node->height = 0;
        node->left = NULL;
        node->right = NULL;
        return node;
    }
    ```

=== "Kotlin"

    ```kotlin title=""
    /* Binary tree node class */
    class TreeNode(val _val: Int) {  // Node value
        var left: TreeNode? = null   // Left child reference
        var right: TreeNode? = null  // Right child reference
    }
    ```

=== "Ruby"

    ```ruby title=""
    ### Binary tree node class ###
    class TreeNode
      attr_accessor :val    # Node value
      attr_accessor :left   # Left child reference
      attr_accessor :right  # Right child reference

      def initialize(val)
        @val = val
      end
    end
    ```

Each node has two references or pointers, one to its <u>left child</u> and one to its <u>right child</u>. The node is their <u>parent</u>. Its left child and all nodes below that child form its <u>left subtree</u>; the <u>right subtree</u> is defined similarly.

**Every non-leaf node has children and at least one nonempty subtree**. In the figure below, if node 2 is the parent, its left and right children are nodes 4 and 5. Its left subtree consists of node 4 and everything below it, and its right subtree consists of node 5 and everything below it.

![Parents, children, and subtrees](binary_tree.assets/binary_tree_definition.png)

## Common terminology

The figure below illustrates common binary-tree terminology.

- <u>Root</u>: the topmost node, which has no parent.
- <u>Leaf</u>: a node with no children; both of its child references point to `None`.
- <u>Edge</u>: a connection between two nodes, represented by a reference or pointer.
- A node's <u>level</u>: levels increase from top to bottom, with the root at level 1.
- A node's <u>degree</u>: its number of children, which is 0, 1, or 2 in a binary tree.
- A tree's <u>height</u>: the number of edges from the root to its farthest leaf.
- A node's <u>depth</u>: the number of edges from the root to that node.
- A node's <u>height</u>: the number of edges from that node to its farthest leaf.

![Common binary-tree terminology](binary_tree.assets/binary_tree_terminology.png)

!!! tip

    Height and depth are usually measured in edges, but some problems or textbooks count nodes instead. Under that convention, add 1 to both height and depth.

## Basic operations

### Initializing a binary tree

As with a linked list, first create the nodes, then connect them with references or pointers.

=== "Python"

    ```python title="binary_tree.py"
    # Initialize a binary tree
    # Initialize nodes
    n1 = TreeNode(val=1)
    n2 = TreeNode(val=2)
    n3 = TreeNode(val=3)
    n4 = TreeNode(val=4)
    n5 = TreeNode(val=5)
    # Connect nodes with references (pointers)
    n1.left = n2
    n1.right = n3
    n2.left = n4
    n2.right = n5
    ```

=== "C++"

    ```cpp title="binary_tree.cpp"
    /* Initialize a binary tree */
    // Initialize nodes
    TreeNode* n1 = new TreeNode(1);
    TreeNode* n2 = new TreeNode(2);
    TreeNode* n3 = new TreeNode(3);
    TreeNode* n4 = new TreeNode(4);
    TreeNode* n5 = new TreeNode(5);
    // Connect nodes with references (pointers)
    n1->left = n2;
    n1->right = n3;
    n2->left = n4;
    n2->right = n5;
    ```

=== "Java"

    ```java title="binary_tree.java"
    // Initialize nodes
    TreeNode n1 = new TreeNode(1);
    TreeNode n2 = new TreeNode(2);
    TreeNode n3 = new TreeNode(3);
    TreeNode n4 = new TreeNode(4);
    TreeNode n5 = new TreeNode(5);
    // Connect nodes with references (pointers)
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;
    ```

=== "C#"

    ```csharp title="binary_tree.cs"
    /* Initialize a binary tree */
    // Initialize nodes
    TreeNode n1 = new(1);
    TreeNode n2 = new(2);
    TreeNode n3 = new(3);
    TreeNode n4 = new(4);
    TreeNode n5 = new(5);
    // Connect nodes with references (pointers)
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;
    ```

=== "Go"

    ```go title="binary_tree.go"
    /* Initialize a binary tree */
    // Initialize nodes
    n1 := NewTreeNode(1)
    n2 := NewTreeNode(2)
    n3 := NewTreeNode(3)
    n4 := NewTreeNode(4)
    n5 := NewTreeNode(5)
    // Connect nodes with references (pointers)
    n1.Left = n2
    n1.Right = n3
    n2.Left = n4
    n2.Right = n5
    ```

=== "Swift"

    ```swift title="binary_tree.swift"
    // Initialize nodes
    let n1 = TreeNode(x: 1)
    let n2 = TreeNode(x: 2)
    let n3 = TreeNode(x: 3)
    let n4 = TreeNode(x: 4)
    let n5 = TreeNode(x: 5)
    // Connect nodes with references (pointers)
    n1.left = n2
    n1.right = n3
    n2.left = n4
    n2.right = n5
    ```

=== "JS"

    ```javascript title="binary_tree.js"
    /* Initialize a binary tree */
    // Initialize nodes
    let n1 = new TreeNode(1),
        n2 = new TreeNode(2),
        n3 = new TreeNode(3),
        n4 = new TreeNode(4),
        n5 = new TreeNode(5);
    // Connect nodes with references (pointers)
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;
    ```

=== "TS"

    ```typescript title="binary_tree.ts"
    /* Initialize a binary tree */
    // Initialize nodes
    let n1 = new TreeNode(1),
        n2 = new TreeNode(2),
        n3 = new TreeNode(3),
        n4 = new TreeNode(4),
        n5 = new TreeNode(5);
    // Connect nodes with references (pointers)
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;
    ```

=== "Dart"

    ```dart title="binary_tree.dart"
    /* Initialize a binary tree */
    // Initialize nodes
    TreeNode n1 = new TreeNode(1);
    TreeNode n2 = new TreeNode(2);
    TreeNode n3 = new TreeNode(3);
    TreeNode n4 = new TreeNode(4);
    TreeNode n5 = new TreeNode(5);
    // Connect nodes with references (pointers)
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;
    ```

=== "Rust"

    ```rust title="binary_tree.rs"
    // Initialize nodes
    let n1 = TreeNode::new(1);
    let n2 = TreeNode::new(2);
    let n3 = TreeNode::new(3);
    let n4 = TreeNode::new(4);
    let n5 = TreeNode::new(5);
    // Connect nodes with references (pointers)
    n1.borrow_mut().left = Some(n2.clone());
    n1.borrow_mut().right = Some(n3);
    n2.borrow_mut().left = Some(n4);
    n2.borrow_mut().right = Some(n5);
    ```

=== "C"

    ```c title="binary_tree.c"
    /* Initialize a binary tree */
    // Initialize nodes
    TreeNode *n1 = newTreeNode(1);
    TreeNode *n2 = newTreeNode(2);
    TreeNode *n3 = newTreeNode(3);
    TreeNode *n4 = newTreeNode(4);
    TreeNode *n5 = newTreeNode(5);
    // Connect nodes with references (pointers)
    n1->left = n2;
    n1->right = n3;
    n2->left = n4;
    n2->right = n5;
    ```

=== "Kotlin"

    ```kotlin title="binary_tree.kt"
    // Initialize nodes
    val n1 = TreeNode(1)
    val n2 = TreeNode(2)
    val n3 = TreeNode(3)
    val n4 = TreeNode(4)
    val n5 = TreeNode(5)
    // Connect nodes with references (pointers)
    n1.left = n2
    n1.right = n3
    n2.left = n4
    n2.right = n5
    ```

=== "Ruby"

    ```ruby title="binary_tree.rb"
    # Initialize a binary tree
    # Initialize nodes
    n1 = TreeNode.new(1)
    n2 = TreeNode.new(2)
    n3 = TreeNode.new(3)
    n4 = TreeNode.new(4)
    n5 = TreeNode.new(5)
    # Connect nodes with references (pointers)
    n1.left = n2
    n1.right = n3
    n2.left = n4
    n2.right = n5
    ```

??? pythontutor "Visualize execution"

    https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Left%20child%20reference%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20Right%20child%20reference%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20binary%20tree%0A%20%20%20%20%23%20Initialize%20nodes%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20Connect%20nodes%20with%20references%20%28pointers%29%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5&cumulative=false&curInstr=3&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false

### Inserting and deleting nodes

As with linked lists, insertion and deletion can be implemented by changing pointers. An example is shown below.

![Inserting and deleting a binary-tree node](binary_tree.assets/binary_tree_add_remove.png)

=== "Python"

    ```python title="binary_tree.py"
    # Insert and delete a node
    p = TreeNode(0)
    # Insert node P between n1 and n2
    n1.left = p
    p.left = n2
    # Delete node P
    n1.left = n2
    ```

=== "C++"

    ```cpp title="binary_tree.cpp"
    /* Insert and delete a node */
    TreeNode* P = new TreeNode(0);
    // Insert node P between n1 and n2
    n1->left = P;
    P->left = n2;
    // Delete node P
    n1->left = n2;
    // Free the memory
    delete P;
    ```

=== "Java"

    ```java title="binary_tree.java"
    TreeNode P = new TreeNode(0);
    // Insert node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Delete node P
    n1.left = n2;
    ```

=== "C#"

    ```csharp title="binary_tree.cs"
    /* Insert and delete a node */
    TreeNode P = new(0);
    // Insert node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Delete node P
    n1.left = n2;
    ```

=== "Go"

    ```go title="binary_tree.go"
    /* Insert and delete a node */
    // Insert node P between n1 and n2
    p := NewTreeNode(0)
    n1.Left = p
    p.Left = n2
    // Delete node P
    n1.Left = n2
    ```

=== "Swift"

    ```swift title="binary_tree.swift"
    let P = TreeNode(x: 0)
    // Insert node P between n1 and n2
    n1.left = P
    P.left = n2
    // Delete node P
    n1.left = n2
    ```

=== "JS"

    ```javascript title="binary_tree.js"
    /* Insert and delete a node */
    let P = new TreeNode(0);
    // Insert node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Delete node P
    n1.left = n2;
    ```

=== "TS"

    ```typescript title="binary_tree.ts"
    /* Insert and delete a node */
    const P = new TreeNode(0);
    // Insert node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Delete node P
    n1.left = n2;
    ```

=== "Dart"

    ```dart title="binary_tree.dart"
    /* Insert and delete a node */
    TreeNode P = new TreeNode(0);
    // Insert node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Delete node P
    n1.left = n2;
    ```

=== "Rust"

    ```rust title="binary_tree.rs"
    let p = TreeNode::new(0);
    // Insert node P between n1 and n2
    n1.borrow_mut().left = Some(p.clone());
    p.borrow_mut().left = Some(n2.clone());
    // Delete node p
    n1.borrow_mut().left = Some(n2);
    ```

=== "C"

    ```c title="binary_tree.c"
    /* Insert and delete a node */
    TreeNode *P = newTreeNode(0);
    // Insert node P between n1 and n2
    n1->left = P;
    P->left = n2;
    // Delete node P
    n1->left = n2;
    // Free the memory
    free(P);
    ```

=== "Kotlin"

    ```kotlin title="binary_tree.kt"
    val P = TreeNode(0)
    // Insert node P between n1 and n2
    n1.left = P
    P.left = n2
    // Delete node P
    n1.left = n2
    ```

=== "Ruby"

    ```ruby title="binary_tree.rb"
    # Insert and delete a node
    _p = TreeNode.new(0)
    # Insert node _p between n1 and n2
    n1.left = _p
    _p.left = n2
    # Delete the node
    n1.left = n2
    ```

??? pythontutor "Visualize execution"

    https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22Binary%20tree%20node%20class%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20Node%20value%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20Left%20child%20reference%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20Right%20child%20reference%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20Initialize%20a%20binary%20tree%0A%20%20%20%20%23%20Initialize%20nodes%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20Connect%20nodes%20with%20references%20%28pointers%29%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20Insert%20and%20delete%20a%20node%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20Insert%20node%20P%20between%20n1%20and%20n2%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20Delete%20node%20P%0A%20%20%20%20n1.left%20%3D%20n2&cumulative=false&curInstr=37&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false

!!! tip

    Inserting a node may change the tree's logical structure, while deleting a node generally means deleting it and all its subtrees. Meaningful insertion and deletion operations therefore usually require several coordinated steps.

## Common types of binary trees

### Perfect binary trees

In a <u>perfect binary tree</u>, every level is completely filled, as shown below. Leaves have degree $0$, and all other nodes have degree $2$. A tree of height $h$ has $2^{h+1} - 1$ nodes, an exponential relationship reminiscent of cell division in nature.

!!! tip

    In Chinese-language resources, a perfect binary tree is often called <u>满二叉树</u>.

![A perfect binary tree](binary_tree.assets/perfect_binary_tree.png)

### Complete binary trees

Only the lowest level of a <u>complete binary tree</u> may be incomplete, and its nodes must fill that level consecutively from left to right, as shown below. Every perfect binary tree is also complete.

![A complete binary tree](binary_tree.assets/complete_binary_tree.png)

### Full binary trees

In a <u>full binary tree</u>, every node other than a leaf has exactly two children, as shown below.

![A full binary tree](binary_tree.assets/full_binary_tree.png)

### Balanced binary trees

In a <u>balanced binary tree</u>, the heights of the left and right subtrees of every node differ by at most 1, as shown below.

![A balanced binary tree](binary_tree.assets/balanced_binary_tree.png)

## Degeneration of a binary tree

The figure below shows ideal and degenerate tree structures. When every level is full, the tree is perfect. When every node lies along one side, the tree degenerates into a linked list.

- A perfect binary tree is the ideal case, making full use of divide and conquer.
- A linked list is the opposite extreme: operations become linear, with time complexity degrading to $O(n)$.

![Best and worst binary-tree structures](binary_tree.assets/binary_tree_best_worst_cases.png)

As the table below shows, quantities such as leaf count, total node count, and height reach their maximum or minimum values in these best and worst structures.

<p align="center"> Table <id> &nbsp; Best and worst binary-tree structures </p>

|                             | Perfect binary tree         | Linked list    |
| --------------------------- | ------------------ | ------- |
| Nodes at level $i$         | $2^{i-1}$          | $1$     |
| Leaves in a tree of height $h$ | $2^h$              | $1$     |
| Total nodes in a tree of height $h$   | $2^{h+1} - 1$      | $h + 1$ |
| Height of a tree with $n$ nodes   | $\log_2 (n+1) - 1$ | $n - 1$ |