Trees

BST Insert

Insert values into a binary search tree by comparing at each node.

Algorithm

The canonical tree is 4(2(1,3),6(5,7)), so this PHP DSA implementation can be compared directly with the rest of the DSA track.

binary search tree Values smaller than a node go left; larger values go right.

Visual walkthrough

BST insertion is a comparison path. The pinned tree 4(2(1,3),6(5,7)) is shown with the inserted value taking its sorted slot.

Step 1 - Start at root

For value 5, compare with 4 first; 5 is larger, so move right.

First comparison: 5 > 4, so the search for the insert slot goes right.insert 54compare26137

Step 2 - Take the left slot under 6

At 6, value 5 is smaller, so it becomes the left child.

Second comparison: 5 < 6, so the open left slot is used.426compare135new7

Step 3 - Canonical tree

The resulting tree is the pinned shape 4(2(1,3),6(5,7)).

Final BST after 5 is present under 6.4261357

Basic Implementation

basic.php 
<?php
class Node {
    public int $value;
    public ?Node $left;
    public ?Node $right;
    public function __construct(int $value, ?Node $left = null, ?Node $right = null) {
        $this->value = $value; $this->left = $left; $this->right = $right;
    }
}
function render_tree(?Node $node): string {
    if ($node === null) return "_";
    if ($node->left === null && $node->right === null) return (string)$node->value;
    return $node->value . "(" . render_tree($node->left) . "," . render_tree($node->right) . ")";
}
function sample_tree(): Node {
    return new Node(4, new Node(2, new Node(1), new Node(3)), new Node(6, new Node(5), new Node(7)));
}
function list_string(array $values): string { return "[" . implode(", ", $values) . "]"; }
function insert_node(?Node $root, int $value): Node { if ($root === null) return new Node($value); if ($value < $root->value) $root->left = insert_node($root->left, $value); else $root->right = insert_node($root->right, $value); return $root; }
$root = null;
foreach ([4, 2, 6, 1, 3, 5, 7] as $value) $root = insert_node($root, $value);
echo render_tree($root) . PHP_EOL;
?>

Complexity

  • Time: O(h) per insert
  • Space: O(n)

Implementation notes

  • Nodes are PHP objects from class Node with typed properties: int $value, ?Node $left, and ?Node $right.
  • The constructor defaults both child links to null, so a new leaf starts with no children.
  • $root starts as null, then each value in [4, 2, 6, 1, 3, 5, 7] is inserted with $root = insert_node($root, $value).
  • insert_node(?Node $root, int $value): Node returns a node every time; the empty-subtree case creates new Node($value).
  • The comparison is $value < $root->value; smaller values recurse into $root->left, while equal-or-larger values use the right branch.
  • Child attachment is recursive assignment: $root->left = insert_node(...) or $root->right = insert_node(...).
  • The trace builds 4, then 4(2,_), 4(2,6), 4(2(1,3),6(5,_)), and finally 4(2(1,3),6(5,7)).
  • render_tree prints _ for a null child and value(left,right) for an internal node, so echo render_tree($root) . PHP_EOL prints 4(2(1,3),6(5,7)).
  • The replay also includes the sorted insert contrast [1, 2, 3, 4], which produces 1(_,2(_,3(_,4))) with height 4 and no rotation step.