Note on A000081 --------------- A000081 is number of rooted trees with n nodes. Claim: This is also the number of ways of arranging n-1 nonoverlapping circles. Proof: The bijection is established by induction on n. First some examples: The root of the tree is indicated by X, the other nodes by small circles o. n=2: the unique tree with 2 nodes: arrangements of 1 circle: o | O X n=3: trees with 3 nodes: arrangements of 2 circles: o | o 2 concentric circles | X o o \ / O O X trees with 4 nodes: arrangements of 3 circle: o | o | o 3 concentric circles | X o \ o o O + 2 concentric circles \/ X o o \ / o big circle containing O O | X o o o \|/ O O O X In general: we build up the isomorphism inductively: Rooted tree Arrangement of circles If a <--> A and b <--> B then a | <--> A inside a circle X and two rooted trees a and b sharing a common root X (for technical reasons X is shown BETWEEN the two trees): ....... ....... .. .. .. .. .. .. .. .. . . . . . tree . tree . . a X b . <--> A next to B . . . . . . . .. .. .. .. .. .. .. .. ....... ....... Exercise: do the mapping for the 9 rooted trees with 5 nodes. Neil Sloane, Oct 29 2004