For a complete binary tree with n nodes, which of the following formulas correctly gives the depth (height in levels) in terms of n, assuming depth grows logarithmically with the number of nodes?

Introduction / Context:
Binary trees are fundamental data structures in computer science, used in searching, sorting, and expression representation. A complete binary tree is a special type in which all levels are completely filled except possibly the last, which is filled from left to right. For such trees, the depth or height grows logarithmically with the number of nodes. This question asks which formula correctly expresses the depth of a complete binary tree in terms of the number of nodes n, using a logarithm with base 2.

Given Data / Assumptions:

• The tree is a complete binary tree, meaning it is as compact as possible.
• n denotes the total number of nodes in the tree.
• Depth (or height) is measured in levels from root to the deepest node.
• The options involve different logarithmic expressions with n + 1 or n − 1 and different bases.
• We assume depth grows on the order of log2 n, as in balanced binary trees.

Concept / Approach:
In a complete binary tree, each level i (starting from 0 at the root) can have at most 2^i nodes. If the tree has height h (with levels 0 to h − 1) and is completely full, the total number of nodes N satisfies N = 2^h − 1. Solving for h gives h = log2(N + 1). For a complete, not necessarily perfect, binary tree, the height is still approximately log2(n + 1), rounding up if needed. Thus, a common formula used in theory questions for the depth of a complete binary tree is Dn = log2(n + 1), capturing this logarithmic relationship with base 2.

Step-by-Step Solution:
Step 1: Start with the formula for a perfect full binary tree: N = 2^h − 1, where N is nodes and h is the number of levels. Step 2: Rearrange the equation to h = log2(N + 1). Step 3: For a complete binary tree, the height is at most this value, and we often approximate the depth as log2(n + 1). Step 4: Examine option A, Dn = log2(n + 1), which matches this derivation and uses base 2. Step 5: Examine option B, which uses log base 1/2. Log base 1/2 decreases as n increases and is not suitable for representing increasing depth. Step 6: Examine option C, which uses log(n + 1) with an unspecified base, making it ambiguous in mathematical contexts where the base matters. Step 7: Examine option D, Dn = log2(n − 1), which does not match the standard derivation from N = 2^h − 1. Step 8: Conclude that Dn = log2(n + 1) is the formula that correctly expresses tree depth in a base 2 logarithmic form.

Verification / Alternative check:
Consider small values: a tree with 1 node (n = 1) has depth 1 level. Evaluating log2(1 + 1) gives log2(2) = 1, which matches. A tree with 3 nodes (n = 3) has depth 2 levels; log2(3 + 1) = log2(4) = 2, again matching. For 7 nodes, depth is 3, and log2(7 + 1) = log2(8) = 3. For non-perfect complete trees, log2(n + 1) provides a close bound, with the actual depth being the ceiling of this value. These examples show that option A fits the expected relationship.

Why Other Options Are Wrong:
Dn = log1/2(n + 1): Using a base less than 1 reverses the direction of growth; depth would appear to decrease as n increases, which contradicts tree behaviour. Dn = log(n + 1) with unspecified base: Without specifying base 2, this is ambiguous; binary trees naturally involve powers of 2, so base 2 is appropriate. Dn = log2(n − 1): Does not match the derivation from N = 2^h − 1 and gives incorrect values for small trees.

Common Pitfalls:
Students sometimes ignore the logarithm base and assume that any log expression is acceptable. In binary tree analysis, base 2 is meaningful because each level doubles the maximum number of nodes. Another pitfall is confusing n − 1 and n + 1 in the formula. Remember that if N = 2^h − 1, then N + 1 = 2^h, leading to h = log2(N + 1). Keeping this derivation in mind helps you avoid sign errors in the expression.

Final Answer:
For a complete binary tree with n nodes, a correct logarithmic expression for depth is Dn = log2(n + 1).