Last Update: 8 December 2010
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Graphs (cont'd) and Rooted Trees
Then T = ({r}, ∅)
is_{def} a rooted tree with root r
Recursive Case:
Let r be a new vertex ≠ each r_{i}
Then the directed graph consisting of r with an edge from r to each r_{i}
Let T_{1}, … , T_{n} be a forest of disjoint trees (rooted at r_{1}, … , r_{n}).
Let r be a vertex ≠ each r_{i}.
Then:
T_{n+1} = ( V_{n+1} = ∪_{i∈{1,…n}}V_{i} ∪
{r},
is a tree rooted at r.
T_{2} = • r_{2}
T_{3} = • r_{3}
T_{4} =
is a rooted tree with root r_{4} by the recursive case
(and T_{1}, T_{2}, T_{3} are subtrees of T_{4})
and if T_{5} =
is a rooted tree with root r_{5},
then T_{7} =
is a rooted tree with root r_{7}
Then here is the "feminist"/"male chauvinist"/neutral terminology:
and, switching metaphors: