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A decision tree represents a deterministic computation of a known Boolean function $f:\{0,1{\}}^n\to \{0,1\}$ by inspecting input bits. Each internal vertex of its binary tree names a coordinate $i$ to query, its two outgoing edges carry answers 0 and 1, and each leaf carries the output 0 or 1. Starting at the root of a tree, read the named bit, follow its answer edge, and repeat until a leaf is reached. Correctness requires the leaf's label to equal $f(x)$ for every input $x$.^[ref]

The next coordinate, or the decision to stop, can depend on earlier answers: this is adaptive querying. The same input always follows the same graph path; adaptivity does not require randomness. A correct leaf must have the same output for every completion of the observed bits. Thus a decision tree can stop as soon as the observations form a query certificate. Querying an already known coordinate adds no information, so we use decision trees without repeated coordinates on a graph path; an efficient tree also removes queries made after the output is already forced.

The motivation is exact computation with shortcuts. If inspecting an input entry is expensive, the question is how much of the input must actually be read. The decision-tree query cost counts these inspections, while local calculations on known values are free in this model; this cost need not equal an implementation's time complexity. Here the Boolean function is specified in advance and must be computed exactly on every input. Decision tree learning instead fits a predictor from data, with a different objective.

The decision tree model computes a function by adaptively querying coordinates of an input. Each internal node specifies a query, its outgoing branches represent possible answers, and a leaf supplies the output. Later queries may depend on earlier answers. The number of queries on the longest root-to-leaf path is the decision-tree depth.^[ref]

For example, consider the Boolean function that returns one if any of its $n$ input bits is one. Querying in order can stop at the first one, but the all-zero input requires all $n$ queries: an unread bit might still change the output. Thus the worst-case query complexity is $n$.

If the input bits are independent and fair, that same procedure uses onlyqueries on average. Worst-case depth and average cost therefore answer different questions, and an average requires a specified input probability distribution.

Harry Buhrman coauthored Complexity Measures and Decision Tree Complexity: A Survey by Harry Buhrman and Ronald de Wolf. The survey relates measures of Boolean-function complexity, including deterministic and randomized decision-tree queries, certificate size and quantum query complexity. These comparisons separate how a function is represented from how much information an algorithm must inspect to compute it.^[ref]

Ronald de Wolf coauthored Complexity Measures and Decision Tree Complexity: A Survey by Harry Buhrman and Ronald de Wolf. Its subject is the relationship between ways of measuring a Boolean function's computational difficulty, particularly query and certificate measures and their quantum counterparts.^[ref]