Metric trees of generalized roundness one

Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable metric trees that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual combinatorial metric (SSTs). Using a simple geometric argument we show how to determine decent upper bounds on the generalized roundness of finite SSTs that depend only on the downward degree sequence of the tree in question. By considering limits it follows that if the downward degree sequence $(d_{0}, d_{1}, d_{2}...)$ of a SST $(T,\rho)$ satisfies $|\{j \, | \, d_{j} > 1 \}| = \aleph_{0}$, then $(T,\rho)$ has generalized roundness one. Included among the trees that satisfy this condition are all complete $n$-ary trees of depth $\infty$ ($n \geq 2$), all $k$-regular trees ($k \geq 3$) and inductive limits of Cantor trees. The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs.Comment: 14 pages, 2 figures, 2 table

Differences in life history traits of related Epilobium species : clonality, seed size and seed number

Small changes in morphology can affect the performance and functions of organisms and hence their ecological success. In modular constructed plants, contrasting growth strategies may be realized by differences in the spatial arrangement and size of shoots. Such differences change the way in which meristems and resources are assigned to various functions during the lifespan of a plant. If such changes include the capacity to spread clonally, sexual reproduction may also be affected. I compare patterns in vegetative growth and sexual reproductive traits in four allopatric species of Epilobium which are sometimes considered as subspecies of a single polymorphic taxon. The four species differ in the location of the buds which annually renew the aerial shoot system. E. dodonaei and E. steveni do not spread clonally and are characterized by a shrub-like habit. E. fleischeri, a species occurring only in the Alps, and E. colchicum, which occurs in the upper region of the Caucasus mountains, both produce buds on horizontal roots or plagiotropic shoots. Both alpine species exhibiting clonal growth have smaller shoots, fewer fruits and smaller seeds than the lowland species. An intraspecific trade-off between seed number per fruit and seed mass is realized. Both alpine species produce mon seeds per fruit at the expense of seed mass. The morphological relationship between the four species and their geographical distribution suggest that clonal growth in E. fleischeri (restricted to the Alps) and E. colchicum (restricted to the Caucasus) is adaptively associated with the stressful conditions of alpine habitats. Our results suggest that clonal growth is not necessarily correlated with reduced reproduction by seeds. The success of plants which are already established may largely depend on clonal spread, but the colonization of new habitats depends on the production of a large number of small seeds with high dispersability