Developing Allometric Equations for Teak Plantations Located in the Coastal Region of Ecuador from Terrestrial Laser Scanning Data

Traditional studies aimed at developing allometric models to estimate dry above-ground biomass (AGB) and other tree-level variables, such as tree stem commercial volume (TSCV) or tree stem volume (TSV), usually involves cutting down the trees. Although this method has low uncertainty, it is quite costly and inefficient since it requires a very time-consuming field work. In order to assist in data collection and processing, remote sensing is allowing the application of non-destructive sampling methods such as that based on terrestrial laser scanning (TLS). In this work, TLS-derived point clouds were used to digitally reconstruct the tree stem of a set of teak trees (Tectona grandis Linn. F.) from 58 circular reference plots of 18 m radius belonging to three different plantations located in the Coastal Region of Ecuador. After manually selecting the appropriate trees from the entire sample, semi-automatic data processing was performed to provide measurements of TSCV and TSV, together with estimates of AGB values at tree level. These observed values were used to develop allometric models, based on diameter at breast height (DBH), total tree height (h), or the metric DBH2 × h, by applying a robust regression method to remove likely outliers. Results showed that the developed allometric models performed reasonably well, especially those based on the metric DBH2 × h, providing low bias estimates and relative RMSE values of 21.60% and 16.41% for TSCV and TSV, respectively. Allometric models only based on tree height were derived from replacing DBH by h in the expression DBH2 x h, according to adjusted expressions depending on DBH classes (ranges of DBH). This finding can facilitate the obtaining of variables such as AGB (carbon stock) and commercial volume of wood over teak plantations in the Coastal Region of Ecuador from only knowing the tree height, constituting a promising method to address large-scale teak plantations monitoring from the canopy height models derived from digital aerial stereophotogrammetry

The forest associated with the record process on a L\'evy tree

We perform a pruning procedure on a L\'evy tree and instead of throwing away the removed sub-tree, we regraft it on a given branch (not related to the L\'evy tree). We prove that the tree constructed by regrafting is distributed as the original L\'evy tree, generalizing a result where only Aldous's tree is considered. As a consequence, we obtain that the quantity which represents in some sense the number of cuts needed to isolate the root of the tree, is distributed as the height of a leaf picked at random in the L\'evy tree

Optimal Binary Search Trees with Near Minimal Height

Suppose we have n keys, n access probabilities for the keys, and n+1 access probabilities for the gaps between the keys. Let h_min(n) be the minimal height of a binary search tree for n keys. We consider the problem to construct an optimal binary search tree with near minimal height, i.e.\ with height h <= h_min(n) + Delta for some fixed Delta. It is shown, that for any fixed Delta optimal binary search trees with near minimal height can be constructed in time O(n^2). This is as fast as in the unrestricted case. So far, the best known algorithms for the construction of height-restricted optimal binary search trees have running time O(L n^2), whereby L is the maximal permitted height. Compared to these algorithms our algorithm is at least faster by a factor of log n, because L is lower bounded by log n

The contact process on finite homogeneous trees revisited

We consider the contact process with infection rate $\lambda$ on $\mathbb{T}_n^d$, the $d$-ary tree of height $n$. We study the extinction time $\tau_{\mathbb{T}_n^d}$, that is, the random time it takes for the infection to disappear when the process is started from full occupancy. We prove two conjectures of Stacey regarding $\tau_{\mathbb{T}_n^d}$. Let $\lambda_2$ denote the upper critical value for the contact process on the infinite $d$-ary tree. First, if $\lambda < \lambda_2$, then $\tau_{\mathbb{T}_n^d}$ divided by the height of the tree converges in probability, as $n \to \infty$, to a positive constant. Second, if $\lambda > \lambda_2$, then $\log \mathbb{E}[\tau_{\mathbb{T}_n^d}]$ divided by the volume of the tree converges in probability to a positive constant, and $\tau_{\mathbb{T}_n^d}/\mathbb{E}[\tau_{\mathbb{T}_n^d}]$ converges in distribution to the exponential distribution of mean 1.Comment: 22 pages, 1 figur