The -tree on vertices is the complete graph . -trees on
vertices are then obtained by joining a new vertex to -cliques the -trees on
vertices in all possible ways.

For example, for ,
the first step adds a new vertex with connecting edge to the 1-cliques (vertices)
of the path graph , giving . Adding another vertex with connecting edges to gives the claw graph and . Similarly, a third iteration gives the star
graph ,
fork graph, and . As can be seen by this construction process, a 1-tree is
simply a normal tree.

The first few 2- and 3-trees are illustrated above.

Gainer-Dewar, A. "-Species and the Enumeration of -Trees." Elec. J. Combin.19, No. 4,
Article P45, 2012.Harary, F. and Palmer, E. M. "On Acyclic
Simplicial Complexes." Mathematika15, 115-122, 1968.Patil,
H. P. "On the Structure of -Trees." J. Combin., Information and System Sci.11,
57-64, 1986.Sloane, N. J. A. Sequences A000055,
A054581, A078792,
A078793, and A201702
in "The On-Line Encyclopedia of Integer Sequences."Xu, S. J.
"The Size of Uniquely Colorable Graphs." J. Combin. Th.50,
319-320, 1990.