(Klarner 1969, Chorneyko and Mohanty 1975). The number of ordered trivalent planted trees with
internal trivalent vertices and leaves other than the plant
is the Catalan number.
Chorneyko, I. Z. and Mohanty, S. G. "On the Enumeration of Certain Sets of Planted Plane Trees." J. Combin. Th. Ser.
B18, 209-221, 1975.Harary, F.; Prins, G.; and Tutte, W. T.
"The Number of Plane Trees." Indag. Math.26, 319-327, 1964.Klarner,
D. A. "A Correspondence Between Sets of Trees." Indag. Math.31,
292-296, 1969.
is defined as a vertex
set 
,
edge set 
, root vertex 
, and order relation 
on 
which satisfies
if 
,
then 
,
where 
is the path length of the graph
path from 
to 
,
,
,
and 
,
then 
internal trivalent vertices and 
leaves other than the plant
is the Catalan number 
.
