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A graceful labelling of a tree T = (V,E), where V is the set of vertices ofthe tree and E is its edge set, is a bijective function f from V to the setconsisting of the numbers 0, 1, ... |E| inclusive, such that if edge uv isassigned the value |f(u)-f(v)| then the edge labels are distinct numbers of theset consisting of the numbers 1, 2, ..., |E| inclusive. A tree is said to be0-roratable if for any of its vertices there is a graceful labelling thatassigns the label 0 to that vertex. A rooted symmetric tree is a tree in whichall vertices at the same level from root vertex have the same degree. It wasknown since 1979 that rooted symmetric trees are graceful and an algebraicdefinition of graceful labelling of this class of trees was found by theauthor. In this paper we prove that rooted symmetric trees with at most 3levels (including root vertex) are 0-rotatable. We also prove that symmetricspider trees with leg length at most 3 and symmetric banana trees, both ofwhich are classes of rooted symmetric trees with 4 levels, are 0-rotatable.Based on these results, we conjecture that all spiders are 0-rotatable andraise the more general question whether all symmetric rooted trees are0-rotatable.

0-rotatability of classes of rooted symmetric trees. Are rooted symmetric trees 0-rotatable?