  In this work we analyze bucket increasing tree families. We introduce two
simple stochastic growth processes, generating random bucket increasing trees
of size $n$, complementing the earlier result of Mahmoud and Smythe for bucket
recursive trees. On the combinatorial side, we define multilabelled
generalizations of the tree families $d$-ary increasing trees and generalized
plane-oriented recursive trees. Additionally, we introduce a clustering process
for ordinary increasing trees and relate it to bucket increasing trees. We
discuss in detail the bucket size two and present a bijection between such
bucket increasing tree families and certain families of graphs called
increasing diamonds, providing an explanation for phenomena observed by Bodini
et al.
