Node Count in Binomial Trees

This question focuses on the size and structure of binomial trees used in option pricing, contrasting a recombining tree with a non-recombining one over the same number of time steps. Candidates must think about how many distinct states the underlying asset can occupy at each time level, and how that translates to the total number of nodes that must be stored or computed. The setup is typical of derivative pricing and risk platforms where tree-based models are used for vanilla and path-dependent options, and where memory, speed, and numerical stability all depend on how the tree grows as the time grid is refined.

To answer it well, you need to reason about combinatorial growth and recognize when a Markov structure allows state-space compression. The question leans on basic counting arguments, growth-rate comparisons, and comfort with orders of magnitude (polynomial versus exponential). Interviewers watch for a clear, structured derivation rather than rote formulas: can the candidate articulate why recombination happens, connect it to path dependence, and then translate that qualitative understanding into a node-count scaling argument relevant for model implementation.

What it tests

The key structure behind binomial trees lies in how paths through the tree either merge (recombine) or remain distinct (non-recombining). In a recombining tree, many different sequences of moves can lead to the same node, so the number of unique nodes at each level grows linearly with the number of steps, resulting in a total count that is quadratic in $N$. In contrast, a non-recombining tree treats every sequence as leading to a unique node, so the number of nodes doubles at each step, leading to exponential growth. This difference arises from whether the process's future evolution depends only on the current state (Markov property and recombination) or on the entire path taken (path-dependence and no recombination). The principle is that recombination compresses the tree by identifying equivalent states, while non-recombination expands it by treating every history as distinct.