Math 2.0

Computer science Definition

Decision tree model

Builds on Rooted treeBoolean function

Definition

Let n≥1n \ge 1 and let f ⁣:{0,1}n→{0,1}f \colon \set{0,1}^n \to \set{0,1} be a Boolean function. A decision tree on nn variables is a finite rooted tree with the following labels.

  1. Each inner vertex is labelled by an index i∈{1,…,n}i \in \set{1, \dots, n} and has exactly two children, reached by edges labelled 00 and 11.
  2. Each leaf is labelled by 00 or 11.

On an input x∈{0,1}nx \in \set{0,1}^n the tree is evaluated by starting at the root. At an inner vertex with label ii one moves along the edge labelled xix_i, and the output is the label of the leaf reached. The tree computes ff if its output is f(x)f(x) for every xx. The cost of the tree is its depth, the largest number of edges from the root to a leaf. The deterministic decision tree complexity D(f)D(f) is the least depth of a decision tree that computes ff.

Remarks

Example: for f(x1,x2)=x1∨x2f(x_1, x_2) = x_1 \vee x_2, let the root query x1x_1. Its 11-child is a leaf labelled 11, and its 00-child queries x2x_2 and has leaves labelled 00 and 11. Both clauses hold, the output is correct on all four inputs, and the depth is 22, so D(f)≤2D(f) \le 2. Non-example: a tree whose root asks whether x1+x2≥1x_1 + x_2 \ge 1 is not a decision tree in this model, because clause 1 allows only queries of one variable. For every ff, D(f)≤nD(f) \le n, since querying all variables suffices. The comparison model is a variant for sorting: inner vertices ask whether ai≤aja_i \le a_j for input numbers a1,…,ana_1, \dots, a_n. A comparison tree that sorts must have at least n!n! leaves, so its depth is at least log⁡2(n!)\log_2(n!), which is of order nlog⁡nn \log n.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Boolean function, definition, no evidence levelBoolean functiondefinitionGraph, definition, no evidence levelGraphdefinitionWalk, definition, no evidence levelWalkdefinitionReachability, definition, no evidence levelReachabilitydefinitionTrail, definition, no evidence levelTraildefinitionConnected graph, definition, no evidence levelConnected graphdefinitionCircuit, definition, no evidence levelCircuitdefinitionCycle, definition, no evidence levelCycledefinitionAcyclic graph, definition, no evidence levelAcyclic graphdefinitionTree, definition, no evidence levelTreedefinitionRooted tree, definition, no evidence levelRooted treedefinitionDecision tree model, definition, no evidence levelDecision tree modeldefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Boolean function definition
  2. Graph definition
  3. Walk definition
  4. Reachability definition
  5. Trail definition
  6. Connected graph definition
  7. Circuit definition
  8. Cycle definition
  9. Acyclic graph definition
  10. Tree definition
  11. Rooted tree definition
  12. Decision tree model definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Decision_tree_model
About this page
TagN70VNYPY3S
URIm2://N70VNYPY3S
Kinddefinition
WritingW0 stub
MSC68Q17, 68Q25, 68Q04
DefinesDecision tree model
UsesRooted tree, Boolean function
Changed3 November 2026

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