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Relative to the resulting DFS edge classification, a tree edge in depth-first search discovers a white vertex and becomes its parent edge. A forward edge in depth-first search goes to a proper descendant but is not that descendant's discovery edge. A back edge in depth-first search goes to an active ancestor, with a self-loop also counted as a back edge when such edges are allowed. A cross edge in depth-first search joins vertices with no ancestor-descendant relationship in the forest, such as different completed branches or different trees.

The definitions refer to directed edges; an undirected graph is normally classified only with tree and back edges, accounting for the reverse copy of a parent edge separately.

False. Take edges $u\to a$ and $c\to u$, and let the outer DFS loop try vertices in order $a,u,c$. The first call finishes $a$ immediately. The next discovers $u$, whose only outgoing edge reaches already black $a$, so its tree contains only $u$. Finally $c$ sees already black $u$.

Vertex $u$ has both an incoming edge from $c$ and an outgoing edge to $a$, yet is the sole vertex of its DFS tree. Nonzero incoming and outgoing degrees do not require either edge to become a tree edge in depth-first search.