A series-reduced tree is a finite tree in which all nodes have vertex degree other than 2 (in other words, no node merely allows a single edge to "pass through"). Series-reduced trees are also called homeomorphically irreducible (Harary and Palmer 1973, pp. 61-62) or topological trees (Bergeron et al. 1998). The numbers of series-reduced trees with 1, 2, ... nodes are 1, 1, 0, 1, 1, 2, 2, 4, 5, 10, 14, ... (OEIS A000014). Harary and Palmer (1973, p. 62, Fig. 3.3.3) illustrate the series-reduced trees on 8 and fewer nodes.
Special cases of series-reduced trees include alkane graphs and star graphs 
on 
nodes.
Series-reduced trees are best known in popular culture due to their appearance in the second blackboard problem in the 1997 film Good Will Hunting, which poses the problem of finding all (10) such trees on 10 nodes.
The numbers of series-reduced planted trees on 
, 2, ... nodes
are 0, 1, 0, 1, 1, 2, 3, 6, 10, 19, 35, ... (OEIS A001678)
and the numbers of series-reduced rooted trees are
1, 1, 0, 2, 2, 4, 6, 12, 20, 39, 71, ... (OEIS A001679).
The process of replacing edges 
and 
incident to a vertex 
of vertex
degree 2 by a single edge 
is known as graph smoothing.
