For a metric continuum X, we consider the hyperspace of subcontinua $C(X)$ of X, with the Hausdorff metric. A Whitney mapping is a continuous function $\mu :C(X)\to [0,\infty )$ such that: (a) for each $p\in X$, $\mu (p)=0$, and (b) if $A,B\in C(X)$ and $A⊊B$, then $\mu (A)<\mu (B)$. The Whitney mapping μ is finitely generated if there exist a finite number of continuous functions $f_1,\dots ,f_n:X\to [0,1]$ such that for each $A\in C(X)$, $\mu (A)=length(f_1(A))+\cdots +length(f_n(A))$. In this paper we study the continua X for which there exist finitely generated Whitney mappings. In particular, when X is a tree, we find relations among the number of necessary mappings to generate a Whitney mapping with: the number of necessary arcs for covering X; the number of end-points of X; the disconnection number of X; the dimension of $C(X)$ and the number n for which X is an n-od.

对于度量连续体X，我们考虑子连续体的超空间$C（X）$的X，与豪斯多夫度量。惠特尼映射是一个连续函数$\mu ：C（X）\to [0，\infty ）$ 这样：（a）每个 $p\in X$， $\mu （p）=0$，以及（b）如果 $\mathrm{一种}，乙\in C（X）$ 和 $\mathrm{一种}⊊乙$， 然后 $\mu （\mathrm{一种}）<\mu （乙）$。如果存在有限数量的连续函数，则将有限生成惠特尼映射μ$F_{\mathrm{1个}}，\dots ，F_{ñ}：X\to [0，\mathrm{1个}]$ 这样每个 $\mathrm{一种}\in C（X）$， $\mu （\mathrm{一种}）=升ËñGŤH（F_{\mathrm{1个}}（\mathrm{一种}））+\cdots +升ËñGŤH（F_{ñ}（\mathrm{一种}））$。在本文中，我们研究了康体X为其中存在有限生成惠特尼映射。特别是，当X是一棵树，我们发现需要映射产生了惠特尼映射的数量之间的关系：覆盖必要的弧的数目X ; X的端点数；X的断开数; 的尺寸$C（X）$和数量Ñ为其X为Ñ -od。