Trees

• How to Fix “Package pgf Error: No Shape Named … Is Known”
  A TikZ “No shape named ... is known” error usually means a coordinate refers to an undefined node. Find the bad reference, check names, and repair path syntax.
• Why Does a Tree with n Vertices Have n − 1 Edges?
  Prove that every finite tree with n vertices has n − 1 edges by induction, using the fact that each nontrivial tree has a leaf.