A spider graph, spider tree, or simply "spider," is a tree with one vertex of degree at least 3 and all others with degree at most 2. The numbers of spiders on , 2, ... nodes are 0, 0, 0, 1, 2, 4, 7, 11, 17, 25, 36, 50, 70, 94, ... (OEIS A004250).
The count of spider trees with nodes is the same as the number of integer partitions of into three or more parts. It also has closed form
Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin.DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Levit, V. E. and Mandrescu, E. "The Independence Polynomial of a Graph--A Survey." In Proceedings of the 1st International Conference on Algebraic Informatics. Held in Thessaloniki, October 20-23, 2005 (Ed. S. Bozapalidis, A. Kalampakas, and G. Rahonis). Thessaloniki, Greece: Aristotle Univ., pp. 233-254, 2005.Sloane, N. J. A. Sequence A004250/M1046 in "The On-Line Encyclopedia of Integer Sequences."
, 2, ... nodes are 0, 0, 0, 1, 2, 4, 7, 11, 17, 25, 36, 50, 70, 94, ... (OEIS A004250). 
of spider trees with 
nodes is the same as the number of integer partitions of 
into three or more parts. It also has closed form 
is the partition function P and 
is the floor function. A generating function for 
is given by 
is a q-Pochhammer symbol. 