Trees And Binary Search Trees

Trees model hierarchical relationships. Binary search trees additionally maintain an ordering invariant between left and right subtrees.

Why This Matters

Trees appear in menus, syntax, filesystems, organizational data, and indexes. Search-tree performance depends on shape; an unbalanced tree can degrade to a list.

Working Model

Traversal can be depth-first or breadth-first. In-order traversal of a valid binary search tree yields sorted keys. Balanced search trees keep height logarithmic, but production PHP code normally uses libraries or database indexes.

Practical Rules

  • State parent, child, root, leaf, depth, and height clearly.
  • Choose traversal order from the required output.
  • Define duplicate-key behavior.
  • Limit depth for untrusted input.
  • Do not hand-build a tree when a database hierarchy query is clearer.

Failure Modes

  • Assuming every binary tree is a binary search tree.
  • Assuming a binary search tree is automatically balanced.
  • Using recursion on attacker-controlled depth without limits.
  • Losing parent-child integrity when moving subtrees.

Verification

  • Test empty and skewed trees.
  • Assert ordering recursively.
  • Compare traversal orders.
  • Measure height as values are inserted.

What You Should Be Able To Do

After this lesson, you should be able to explain tree terminology, traversal, binary-search-tree ordering, and balancing tradeoffs, choose a suitable approach for a real PHP project, and verify the result instead of relying on assumptions.

Practice

Practice: Traverse A Menu Tree

Render a nested menu in depth-first order with a maximum depth.

Your answer must:

  • state the intended outcome;
  • show the commands, data flow, or implementation shape;
  • identify at least one unsafe alternative;
  • explain how the result will be verified.
Show solution

Visit each node, emit it with its depth, recurse into children only below the limit, and reject or mark cycles if IDs repeat.

The important part is not memorising one command or vendor screen. The solution makes the invariant, failure behavior, and verification evidence explicit.

Practice: Validate A Binary Search Tree

Design validation for the whole tree rather than only parent-child pairs.

Your answer must:

  • state the intended outcome;
  • show the commands, data flow, or implementation shape;
  • identify at least one unsafe alternative;
  • explain how the result will be verified.
Show solution

Carry permitted lower and upper bounds through recursion. Every node must satisfy the full ancestor-derived interval, not only compare with its direct parent.

The important part is not memorising one command or vendor screen. The solution makes the invariant, failure behavior, and verification evidence explicit.

Practice: Explain Tree Degeneration

Insert already-sorted values into a naive binary search tree and analyze the result.

Your answer must:

  • state the intended outcome;
  • show the commands, data flow, or implementation shape;
  • identify at least one unsafe alternative;
  • explain how the result will be verified.
Show solution

Each value becomes the right child of the previous node, height becomes linear, and lookup loses logarithmic behavior. Use balancing or another structure.

The important part is not memorising one command or vendor screen. The solution makes the invariant, failure behavior, and verification evidence explicit.