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Lesson 3 of 8

Count calls separately from stack depth

Distinguish a chain, a branching tree, and divide-and-conquer work.

Understand the idea

Maximum simultaneously active calls determine stack depth. Total calls and work inside each call determine running time. A binary decision tree of depth n can have 2^n leaves while using only O(n) active stack frames during depth-first traversal.

Trace a small example

Naive Fibonacci recomputes overlapping states and has exponential work, despite linear maximum depth. Splitting an array in half and merging the results performs O(n) work per level across O(log n) levels, yielding O(n log n). Recursive binary search keeps only one half and takes O(log n) calls.

Conditions and cost

Printing every length-n binary string alone requires Ω(n × 2^n) output work. Small depth does not imply fast execution. Very deep linear recursion may exhaust the process stack even when its arithmetic is safe.

Growth explorer

Idealized operation counts show growth. Actual runtime also depends on the implementation and machine.

O(1)1
O(log n)7
O(n)100
O(n log n)665
O(n²)10,000

Bar lengths use a logarithmic scale. Logarithms use base 2.

Guided simulation · Change the inputs to test your understanding.

Pause & predict

A traversal has 2^n leaves but follows one branch at a time. What can its stack depth be?

Choose an answer
Try it yourself

Turn the idea into code

Compare recursive binary search and a full take-or-skip subset search on n positions.

Give me a hint

Ask how many child branches are explored from each state.

Show the explained solution

Binary search explores one half per step, giving O(log n) calls. Subset search explores both choices and has 2^n leaves, with O(n) maximum depth.

Your notes

Ready for the next idea?

Mark complete when you can explain it yourself.

From a problem to a programIdentify the input, output, and constraints, then translate a small plan into C++.Variables, types, and remembered valuesChoose a representation, initialize it, and understand what assignment really changes.Arithmetic without surprisesPredict division and remainder, widen before multiplication, and format decimal answers.Conditions and logical decisionsBuild exclusive categories, combine comparisons, and protect operations with short-circuit logic.Optional: other ways to chooseRecognize switch for exact choices and the conditional operator for choosing a value.For loops: make repetition preciseTrace initialization, condition, body, and update—and avoid an extra iteration.While, do-while, and stopping rulesChoose a loop from its stopping condition and read the exact input format.Break, continue, and early answersDistinguish leaving a loop from skipping one iteration, without losing required input.Five patterns worth understandingUse sums, counts, best-so-far candidates, products, and flags to process a stream.Nested loops: rows and columnsTranslate a shape into two loop roles and count the total work.Take a number apart, one digit at a timeUse remainder and division to process decimal digits, including zero.Multiple cases and a debugging mindsetCombine the patterns, isolate each case, and test the boundaries that can break your idea.Arrays: give every value a placeUnderstand indices, initialization, capacity, active length, and the first safe array loop.Scan, count, search, and compareDistinguish values from positions and preserve the exact tie rule a problem asks for.Change a sequence without losing itUpdate, copy, compare, reverse, and inspect neighbours using the correct bounds.Matrices: think in two coordinatesTraverse tables, total rows and columns, inspect diagonals, and protect grid neighbours.Characters become stringsSeparate characters from numbers, then safely inspect and edit managed text.Read the text the problem actually givesChoose words or full lines, preserve empty cases, and reset only the state that needs it.Scan text: count, filter, transformBuild a result from matching characters without losing leading zeros or skipping shifted text.Join, compare, and find textLearn content comparison, lexicographical order, substring counts, and the not-found sentinel.