zyB_17_1
Suppose a recursive function's runtime is
T(n−1)+Θ(n){"version":"1.1","math":"T(n-1) + \Theta(n)"}
How many levels will the recursion tree have?

Question

zyB_17_1
Suppose a recursive function's runtime is 
T(n−1)+Θ(n){"version":"1.1","math":"T(n-1) + \Theta(n)"}
How many levels will the recursion tree have?

BlackTom AI · Accepted Answer

We start by examining the given recurrence: T(n) = T(n-1) + Θ(n). This means each recursive call reduces the problem size by 1, and at each level the non-recursive work is on the order of the current n (roughly Θ(n), Θ(n-1), and so on as we go deeper).

Option 1: logn. This would imply a recursion depth proportional to the logarithm of n, which would occur if we halved the problem size at each step (e.g., T(n) = T(n/2) + ...). Since the size decreases by exactly 1 each level, the depth is not logarithmic but linear in n.

Option 2: nlogn. This is a description typically associated with total cost across levels when summing a Θ(n) cost across Θ(log n) levels, or other patterns, but it conflates depth with total cost. The depth itself is not nlogn; rather, that would be a possible overall running time in some scenarios, not the number of levels.

Option 3: n^2. This would suggest a quadratic-depth structure, which would require the depth to scale as n times n, which is not the case here since we only reduce by 1 each step, giving at most n levels.

Option 4: n. Because each recursive call reduces the problem size by 1 and we continue until a base case (e.g., n reaches 0 or 1), the recursion tree has one level per value of the problem size from n down to 1. Thus, the depth (number of levels) grows linearly with n, i.e., O(n).

In summary, the depth of the recursion tree is linear in n, which aligns with the option describing a depth of n. The other options mischaracterize the depth growth by implying logarithmic, quadratic, or combined total-cost patterns that do not reflect the single-step decrement per level in this recurrence.

zyB_17_1 Suppose a recursive function's runtime is T(n−1)+Θ(n){"version":"1.1","math":"T(n-1) + \Theta(n)"} How many levels will the recursion tree have?

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