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When exploring the characteristics of a full ternary tree—a type of tree commonly studied in discrete mathematics—we must first understand the concept of tree height. The height of a tree is indicative of its deepest layer, which directly influences its overall structure. In a full ternary tree, every non-leaf node expands into exactly three children, leading to a broad and well-balanced formation.

Determining the tree height is straightforward: starting from the root (which is considered level 0), each step down towards the leaves counts as a progression in height. For instance, if a tree's height is 5, it means there are 5 levels beneath the root—each level expanding thrice as much as its predecessor. It also provides insight into potential growth, as trees of greater height have expanded capacity for additional nodes and, subsequently, leaves.

The number of leaves in a full ternary tree is a fundamental concept in discrete mathematics as well, especially when analyzing tree algorithms or investigating tree dynamics. Since leaves are the terminal nodes of a tree, without any children, they form the tree's outermost level. In the context of a full ternary tree, where non-leaf nodes have exactly three children, the leaves represent the culmination of multiplication at each level.

Using the formula of maximum nodes for a given level, \(3^n\), where \(n\) signifies the level (excluding the root), we can calculate the total potential leaves at a tree's final layer. This calculation gives us a clear visual representation of tree growth at each stage, providing vital data on its capacity to expand and its potential impact within a system.

Discrete mathematics is the cornerstone of understanding structures like trees in computer science and mathematics. It encompasses a wide array of topics, from graph theory and combinatorics to logic and set theory. When applied to the study of trees, discrete mathematics allows for quantification and algorithmic analysis of structures.

In addition to offering methods to represent and calculate properties of trees, such as height and the number of leaves, it also facilitates an in-depth understanding of the inherently discrete nature of these constructs. By employing this branch of mathematics, students can tackle complex problems by breaking them into manageable, discrete units. It's this very approach that makes the comprehension of full ternary trees—and many other mathematical structures—not just possible, but approachable for learners at all levels.