Studies on the effect of nanofertilizers (NF) in physiological performance of plants is scarce, especially that related to substances encapsulated into silicon dioxide (SiO2) nanoparticles in cocoa plants. The effect of foliar application of SiO2-NF on nutrient contents, gas exchange, photochemical activity, photosynthetic pigments, total soluble protein (TSP), photosynthetic nitrogen use efficiency (PNUE), and growth in seedlings of two cocoa clones (OC-61 and BR-05) in a greenhouse was assessed. Spraying with SiO2-NF increased net photosynthetic rate (A) by 16 and 60% and electron transport rate (J) by 52 and 162% in clones OC-61 and BR-05, respectively, without changes in photosynthetic pigment concentration in either clone. The SiO2-NF caused a decrease of 37 and 22% in stomatal conductance in OC-61 and BR-05, respectively; a similar trend was observed in transpiration rate, causing an increase of 42 and 100% in water use efficiency in OC-61 and BR-05, respectively. In both clones, diameter of graft increased on average 28% with SiO2-NF. Higher photosynthetic capacity was related to an increase in leaf N, P, and TSP. A significant reduction in PNUE (A/N ratio) was found in OC-61, whereas in BR-05 PNUE increased after spraying with SiO2-NF. Overall, spraying with SiO2-NF had a positive effect on photosynthetic processes in both cocoa clones, associated with an increase in nutrients content, which translated into improved growth. A differential physiological response to spraying with SiO2-NF between clones was also found, with BR-05 being the clone with a better physiological response during the establishment and development stages.

Denote by Sn the set of all distinct rooted trees with n labeled vertices. Define τn as the total height of a tree chosen at random in the set Sn, assuming that all the possible nn–1 choices are equally probable. The total height of a tree is defined as the sum of the heights of its vertices. The height of a vertex in a rooted tree is the distance from the vertex to the root of the tree, that is, the number of edges in the path from the vertex to the root. This paper is concerned with the distribution and the moments of τn and their asymptotic behavior as n → ∞.