Theory

tree

• an undirected graph that is connected and has no cycles
• smallest tree is a single isolated vertex

rooted trees

• any tree becomes a rooted tree by distinguishing any of its vertices as root and directing all edges away from the root

  with no order among children

  • we simply have a root and all edges directed away from it

  with order on children

  • in addition, we distinguish an order among the children

non-rooted trees

• any rooted tree becomes a non-rooted tree if we forget the distinguished vertex and the parent-child direction of edges

binary trees

• in addition, we sometimes distinguish between a node with only a left child and a node with only a right child (this is the case for binary search trees, for example)

an undirected graph:

• the graph is a tree
• the graph is connected and has at most n-1 edges (where n is the number of vertices)
• the graph has no cycles and has at least n-1 edges
• the graph is connected and minimal (if we remove any edge, it becomes non-connected)
• the graph has no cycles and is maximal (if we add any edge, it closes a cycle)
• for any pair of vertices, there is a unique path connecting them

minimum spanning tree