1 of a vertex of a graph. The number of edges incident with the vertex, i.e. that emanate from that vertex. In a directed graph, the indegree is the number of edges entering a vertex while the outdegree is the number leaving a vertex.
2 of a node in a tree. The number of children of that node, i.e. the number of subtrees rooted at that node. More correctly, this is the outdegree of the node.
3 of a tree. The maximum degree of all the nodes in the tree.
4 of a polynomial. See polynomial.
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