Day 17: mat385

Today:

• Collect homework 5.1

• Section 5.2: Trees and their representations
  • Wffs as trees, and traversal.
  • Unary operations make the "left" rule make sense for infix notation.
  • The proof that a tree with n nodes has n-1 arcs (Example 29, p. 380) is a corollary of Euler's formula for planar graphs.

• Questions on Homework for 5.2?
  • Question from last time (re 5.1):

    where in the proof of Euler's formula (by induction on arcs) was simplicity required?
    (Was it merely accidental that the formula worked for the non-simple graph of figure 5.17, p. 350?)

• Section 5.3: Decision Trees

Next time: section 6.2: Euler Paths and Hamiltonian Circuits

Links:

• A friendly glossary of graph terminology

• NIST (the National Institute of Standards and Technology) defines it as "A connected, undirected, acyclic graph". Our book does not require the "undirected" part, and many sources assume that a tree is directed! This does, unfortunately, add to the confusion.... An ordered tree is one where the children are numbered (there is a left to right), which is important for tree traversal.

• NIST (the National Institute of Standards and Technology) Dictionary of Algorithms and Data Structures

• No need to restrict Euler's formula to simple graphs! If you find a theorem with restrictions, ask yourself why the restrictions are there! (Now: is connectedness important?)