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In-order traversal is a critical concept in understanding binary search tree operations. It is a method of traversing a binary search tree where one visits the nodes in a non-decreasing order, which for binary search trees, results in the nodes being visited in lexicographic order. During in-order traversal, the algorithm recursively visits the left subtree, processes the current node, and finally visits the right subtree.

When implemented correctly, in-order traversal ensures that if you have a binary search tree of English words, for instance, they will be printed alphabetically. This property is particularly useful for tasks like sorting data and obtaining an ordered list from a binary search tree. The process can be visualized as such:

• Visit all nodes in the left subtree
• Visit the current node (process, or in this case, print the value)
• Visit all nodes in the right subtree

By adhering to this method, the algorithm respects the inherent ordered structure of a binary search tree.

Algorithm design is a meticulous process that involves creating a step-by-step solution to solve a specific problem or perform a computation. In the case of binary tree traversal, the algorithm must adhere to certain rules to achieve an efficient and correct in-order traversal. Effective algorithm design necessitates understanding the problem domain—in this case, the properties of binary search trees—and designing an algorithm that exploits these properties to achieve a desired outcome.

For our lexicographic printing task, the algorithm is designed to first check if the tree is empty, avoiding unnecessary operations on a null structure. Then it follows the in-order traversal pattern to visit each node in lexicographic order. The simplicity of this algorithm lies in its recursive structure, which elegantly breaks down the traversal process into smaller, identical sub-problems. This recursive approach simplifies the design and implementation, making the code easier to understand and maintain.

A recursive function is a function that calls itself in order to solve a problem. Recursive functions are particularly useful when a problem can be broken down into smaller, more manageable sub-problems of the same form as the original. It's a cornerstone in the algorithm design for various tree traversals, including in-order traversal.

The `in_order_traversal` function in our algorithm is a perfect example of a recursive function; it calls itself to traverse the left and right subtrees of a given node. Each call to the function represents a step closer to the base case, which, in the context of the in-order traversal, is reaching a node that does not have any children (a leaf node) or is `None`. Once the base case is reached, the function returns, unwinding the recursive calls and continuing the traversal. Recursion can seem tricky at first, but understanding that each call is independent and maintains its own state is crucial for grasping how the in-order traversal works.

Lexicographic order is a term borrowed from lexicography (the making of dictionaries) and is used to describe a sequence that is sorted in the same order as a standard dictionary for a particular language. When data is sorted lexicographically, it follows the order defined by the lexicographic conventions of the language it represents—typically alphabetical order for strings or numerical order for numbers.

In the context of traversing a binary search tree lexicographically, this means that if our binary search tree stores words, an in-order traversal would visit these words as if one is browsing through a dictionary, from 'aardvark' to 'zygote'. This ordering is naturally achieved with a binary search tree due to its structure, where all elements to the left of a node are lexicographically smaller, and all to the right are lexicographically greater. Thus, the in-order traversal algorithm prints or lists the tree's contents following this strict lexicographic rule without additional sorting.