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# 7.1 Binary Tree

A binary tree is a non-linear data structure that models the hierarchical relationship between "ancestors" and "descendants" and embodies a divide-and-conquer pattern in which each split branches into two. Similar to a linked list, the basic unit of a binary tree is a node, and each node contains a value, a reference to its left child node, and a reference to its right child node.

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby

    class TreeNode:
        """Binary tree node"""
        def __init__(self, val: int):
            self.val: int = val                # Node value
            self.left: TreeNode | None = None  # Reference to left child node
            self.right: TreeNode | None = None # Reference to right child node

    /* Binary tree node */
    struct TreeNode {
        int val;          // Node value
        TreeNode *left;   // Pointer to left child node
        TreeNode *right;  // Pointer to right child node
        TreeNode(int x) : val(x), left(nullptr), right(nullptr) {}
    };

    /* Binary tree node */
    class TreeNode {
        int val;         // Node value
        TreeNode left;   // Reference to left child node
        TreeNode right;  // Reference to right child node
        TreeNode(int x) { val = x; }
    }

    /* Binary tree node */
    class TreeNode(int? x) {
        public int? val = x;    // Node value
        public TreeNode? left;  // Reference to left child node
        public TreeNode? right; // Reference to right child node
    }

    /* Binary tree node */
    type TreeNode struct {
        Val   int
        Left  *TreeNode
        Right *TreeNode
    }
    /* Constructor */
    func NewTreeNode(v int) *TreeNode {
        return &TreeNode{
            Left:  nil, // Pointer to left child node
            Right: nil, // Pointer to right child node
            Val:   v,   // Node value
        }
    }

    /* Binary tree node */
    class TreeNode {
        var val: Int // Node value
        var left: TreeNode? // Reference to left child node
        var right: TreeNode? // Reference to right child node

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

    /* Binary tree node */
    class TreeNode {
        val; // Node value
        left; // Pointer to left child node
        right; // Pointer to right child node
        constructor(val, left, right) {
            this.val = val === undefined ? 0 : val;
            this.left = left === undefined ? null : left;
            this.right = right === undefined ? null : right;
        }
    }

    /* Binary tree node */
    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; // Reference to left child node
            this.right = right === undefined ? null : right; // Reference to right child node
        }
    }

    /* Binary tree node */
    class TreeNode {
      int val;         // Node value
      TreeNode? left;  // Reference to left child node
      TreeNode? right; // Reference to right child node
      TreeNode(this.val, [this.left, this.right]);
    }

    use std::rc::Rc;
    use std::cell::RefCell;

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

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

    /* Binary tree node */
    typedef struct TreeNode {
        int val;                // Node value
        int height;             // Node height
        struct TreeNode *left;  // Pointer to left child node
        struct TreeNode *right; // Pointer to right child node
    } 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;
    }

    /* Binary tree node */
    class TreeNode(val _val: Int) {  // Node value
        var left: TreeNode? = null   // Reference to left child node
        var right: TreeNode? = null  // Reference to right child node
    }

    ### Binary tree node class ###
    class TreeNode
      attr_accessor :val    # Node value
      attr_accessor :left   # Reference to left child node
      attr_accessor :right  # Reference to right child node

      def initialize(val)
        @val = val
      end
    end

Each node has two references (pointers), pointing respectively to the left-child node and right-child node. This node is called the parent node of these two child nodes. When given a node of a binary tree, we call the tree formed by this node's left child and all nodes below it the left subtree of this node. Similarly, the right subtree can be defined.

In a binary tree, every non-leaf node has child nodes and therefore non-empty subtrees. As shown in Figure 7-1, if "Node 2" is regarded as a parent node, its left and right child nodes are "Node 4" and "Node 5" respectively. The left subtree is formed by "Node 4" and all nodes beneath it, while the right subtree is formed by "Node 5" and all nodes beneath it.

Figure 7-1 Parent Node, child Node, subtree

## 7.1.1 Common Terminology of Binary Trees

The commonly used terminology of binary trees is shown in Figure 7-2.

- Root node: The node at the top level of a binary tree, which does not have a parent node.
- Leaf node: A node that does not have any child nodes, with both of its pointers pointing to `None`.
- Edge: A line segment that connects two nodes, representing a reference (pointer) between the nodes.
- The level of a node: It increases from top to bottom, with the root node being at level 1.
- The degree of a node: The number of child nodes that a node has. In a binary tree, the degree can be 0, 1, or 2.
- The height of a binary tree: The number of edges from the root node to the farthest leaf node.
- The depth of a node: The number of edges from the root node to the node.
- The height of a node: The number of edges from the farthest leaf node to the node.

Figure 7-2 Common Terminology of Binary Trees

Tip

We usually define "height" and "depth" as the number of edges traversed, but some textbooks and problem statements define them as the number of nodes on the path. In that case, both values are larger by 1.

## 7.1.2 Basic Operations of Binary Trees

### 1. Initializing a Binary Tree

Similar to a linked list, the initialization of a binary tree involves first creating the nodes and then establishing the references (pointers) between them.

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby
binary_tree.py

    # Initializing a binary tree
    # Initializing nodes
    n1 = TreeNode(val=1)
    n2 = TreeNode(val=2)
    n3 = TreeNode(val=3)
    n4 = TreeNode(val=4)
    n5 = TreeNode(val=5)
    # Linking references (pointers) between nodes
    n1.left = n2
    n1.right = n3
    n2.left = n4
    n2.right = n5

binary_tree.cpp

    /* Initializing a binary tree */
    // Initializing 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);
    // Linking references (pointers) between nodes
    n1->left = n2;
    n1->right = n3;
    n2->left = n4;
    n2->right = n5;

binary_tree.java

    // Initializing 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);
    // Linking references (pointers) between nodes
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;

binary_tree.cs

    /* Initializing a binary tree */
    // Initializing nodes
    TreeNode n1 = new(1);
    TreeNode n2 = new(2);
    TreeNode n3 = new(3);
    TreeNode n4 = new(4);
    TreeNode n5 = new(5);
    // Linking references (pointers) between nodes
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;

binary_tree.go

    /* Initializing a binary tree */
    // Initializing nodes
    n1 := NewTreeNode(1)
    n2 := NewTreeNode(2)
    n3 := NewTreeNode(3)
    n4 := NewTreeNode(4)
    n5 := NewTreeNode(5)
    // Linking references (pointers) between nodes
    n1.Left = n2
    n1.Right = n3
    n2.Left = n4
    n2.Right = n5

binary_tree.swift

    // Initializing 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)
    // Linking references (pointers) between nodes
    n1.left = n2
    n1.right = n3
    n2.left = n4
    n2.right = n5

binary_tree.js

    /* Initializing a binary tree */
    // Initializing nodes
    let n1 = new TreeNode(1),
        n2 = new TreeNode(2),
        n3 = new TreeNode(3),
        n4 = new TreeNode(4),
        n5 = new TreeNode(5);
    // Linking references (pointers) between nodes
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;

binary_tree.ts

    /* Initializing a binary tree */
    // Initializing nodes
    let n1 = new TreeNode(1),
        n2 = new TreeNode(2),
        n3 = new TreeNode(3),
        n4 = new TreeNode(4),
        n5 = new TreeNode(5);
    // Linking references (pointers) between nodes
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;

binary_tree.dart

    /* Initializing a binary tree */
    // Initializing 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);
    // Linking references (pointers) between nodes
    n1.left = n2;
    n1.right = n3;
    n2.left = n4;
    n2.right = n5;

binary_tree.rs

    // Initializing 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);
    // Linking references (pointers) between nodes
    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);

binary_tree.c

    /* Initializing a binary tree */
    // Initializing nodes
    TreeNode *n1 = newTreeNode(1);
    TreeNode *n2 = newTreeNode(2);
    TreeNode *n3 = newTreeNode(3);
    TreeNode *n4 = newTreeNode(4);
    TreeNode *n5 = newTreeNode(5);
    // Linking references (pointers) between nodes
    n1->left = n2;
    n1->right = n3;
    n2->left = n4;
    n2->right = n5;

binary_tree.kt

    // Initializing nodes
    val n1 = TreeNode(1)
    val n2 = TreeNode(2)
    val n3 = TreeNode(3)
    val n4 = TreeNode(4)
    val n5 = TreeNode(5)
    // Linking references (pointers) between nodes
    n1.left = n2
    n1.right = n3
    n2.left = n4
    n2.right = n5

binary_tree.rb

    # Initializing a binary tree
    # Initializing nodes
    n1 = TreeNode.new(1)
    n2 = TreeNode.new(2)
    n3 = TreeNode.new(3)
    n4 = TreeNode.new(4)
    n5 = TreeNode.new(5)
    # Linking references (pointers) between nodes
    n1.left = n2
    n1.right = n3
    n2.left = n4
    n2.right = n5

Code Visualization

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### 2. Inserting and Removing Nodes

Similar to a linked list, inserting and removing nodes in a binary tree can be achieved by modifying pointers. Figure 7-3 provides an example.

Figure 7-3 Inserting and removing nodes in a binary tree

Python C++Java C#Go Swift JS TS Dart Rust C Kotlin Ruby
binary_tree.py

    # Inserting and removing nodes
    p = TreeNode(0)
    # Inserting node P between n1 -> n2
    n1.left = p
    p.left = n2
    # Removing node P
    n1.left = n2

binary_tree.cpp

    /* Inserting and removing nodes */
    TreeNode* P = new TreeNode(0);
    // Inserting node P between n1 and n2
    n1->left = P;
    P->left = n2;
    // Removing node P
    n1->left = n2;

binary_tree.java

    TreeNode P = new TreeNode(0);
    // Inserting node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Removing node P
    n1.left = n2;

binary_tree.cs

    /* Inserting and removing nodes */
    TreeNode P = new(0);
    // Inserting node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Removing node P
    n1.left = n2;

binary_tree.go

    /* Inserting and removing nodes */
    // Inserting node P between n1 and n2
    p := NewTreeNode(0)
    n1.Left = p
    p.Left = n2
    // Removing node P
    n1.Left = n2

binary_tree.swift

    let P = TreeNode(x: 0)
    // Inserting node P between n1 and n2
    n1.left = P
    P.left = n2
    // Removing node P
    n1.left = n2

binary_tree.js

    /* Inserting and removing nodes */
    let P = new TreeNode(0);
    // Inserting node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Removing node P
    n1.left = n2;

binary_tree.ts

    /* Inserting and removing nodes */
    const P = new TreeNode(0);
    // Inserting node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Removing node P
    n1.left = n2;

binary_tree.dart

    /* Inserting and removing nodes */
    TreeNode P = new TreeNode(0);
    // Inserting node P between n1 and n2
    n1.left = P;
    P.left = n2;
    // Removing node P
    n1.left = n2;

binary_tree.rs

    let p = TreeNode::new(0);
    // Inserting node P between n1 and n2
    n1.borrow_mut().left = Some(p.clone());
    p.borrow_mut().left = Some(n2.clone());
    // Removing node P
    n1.borrow_mut().left = Some(n2);

binary_tree.c

    /* Inserting and removing nodes */
    TreeNode *P = newTreeNode(0);
    // Inserting node P between n1 and n2
    n1->left = P;
    P->left = n2;
    // Removing node P
    n1->left = n2;

binary_tree.kt

    val P = TreeNode(0)
    // Inserting node P between n1 and n2
    n1.left = P
    P.left = n2
    // Removing node P
    n1.left = n2

binary_tree.rb

    # Inserting and removing nodes
    _p = TreeNode.new(0)
    # Inserting node _p between n1 and n2
    n1.left = _p
    _p.left = n2
    # Removing node _p
    n1.left = n2

Code Visualization

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Tip

Keep in mind that inserting a node can alter the original logical structure of a binary tree, while deleting a node usually entails removing that node together with its entire subtree. In practice, insertion and deletion in binary trees are therefore typically implemented as coordinated sequences of operations to achieve a meaningful result.

## 7.1.3 Common Types of Binary Trees

### 1. Perfect Binary Tree

As shown in Figure 7-4, a perfect binary tree has every level completely filled. In a perfect binary tree, leaf nodes have a degree of \(0\), while all other nodes have a degree of \(2\). If the tree height is \(h\), the total number of nodes is \(2^{h+1} - 1\), following a standard exponential pattern that mirrors the common phenomenon of cell division in nature.

Tip

Please note that in the Chinese community, a perfect binary tree is often referred to as a full binary tree.

Figure 7-4 Perfect binary tree

### 2. Complete Binary Tree

As shown in Figure 7-5, a complete binary tree only allows the bottom level to be incompletely filled, and the nodes at the bottom level must be filled continuously from left to right. Note that a perfect binary tree is also a complete binary tree.

Figure 7-5 Complete binary tree

### 3. Full Binary Tree

As shown in Figure 7-6, in a full binary tree, all nodes except leaf nodes have two child nodes.

Figure 7-6 Full binary tree

### 4. Balanced Binary Tree

As shown in Figure 7-7, in a balanced binary tree, the absolute difference between the height of the left and right subtrees of any node does not exceed 1.

Figure 7-7 Balanced binary tree

## 7.1.4 Degeneration of Binary Trees

Figure 7-8 contrasts the ideal and degenerate structures of binary trees. When every level is filled, the tree becomes a "perfect binary tree"; when all nodes skew to one side, the binary tree degenerates into a "linked list".

- A perfect binary tree is the ideal case, fully leveraging the divide-and-conquer advantages of binary trees.
- A linked list represents the other extreme, where all operations become linear operations with time complexity degrading to \(O(n)\).

Figure 7-8 The Best and Worst Structures of Binary Trees

As shown in Table 7-1, in the best and worst structures, the binary tree achieves either maximum or minimum values for leaf node count, total number of nodes, and height.

Table 7-1 The Best and Worst Structures of Binary Trees

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

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