Let $G$ be a simple graph and $\Delta(G)$ denote the maximum degree of $G$. A harmonious colouring of $G$ is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number $h(G)$ is the least number of colours in such a colouring. In this paper it is shown that if $T$ is a tree of order $n$ and $\Delta(T)\geq\frac{n}{2}$, then there exists a harmonious colouring of $T$ with $\Delta(T)+1$ colours such that every colour is used at most twice. Thus $h(T)=\Delta(T)+1$. Moreover, we prove that if $T$ is a tree of order $n$ and $\Delta(T) \le \Big\lceil\frac{n}{2}\Big\rceil$, then there exists a harmonious colouring of $T$ with $\Big\lceil \frac{n}{2}\Big \rceil +1$ colours such that every colour is used at most twice. Thus $h(T)\leq \Big\lceil \frac{n}{2} \Big\rceil +1$.

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In this paper, we introduce the notion of the equivalence relation, called n-isoclinism, between crossed modules of groups, and give some basic properties of this notion. In particular, we obtain some criteria under which crossed modules are n-isoclinic. Also, we present the notion of n-stem crossed module and, under some conditions, determine them within an n-isoclinism class.

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In this article we investigate the interplay between stem covers, the Schur multiplier of Leibniz crossed modules and the non-abelian exterior product of Leibniz algebras. In concrete, we obtain a six-term exact sequence associated to a central extension of Leibniz crossed modules, which is useful to characterize stem covers. We show the existence of stem covers and determine the structure of all stem covers of Leibniz crossed modules. Also, we give the connection between the stem cover of a Lie crossed module in the categories of Lie and Leibniz crossed modules respectively.