Zernike radial polynomials (ZRP) play a significant role in application areas such as optics design, imaging systems, and image processing systems. Currently, there are two kinds of numerical schemes for computing the ZRP automatically with computer programs: one is based on the definition in which the factorial operations may lead to the overflow problem and the high order derivatives are troublesome, and the other is based on recursion which is either unstable or with high computational complexity. In this paper, our emphasis is focused on exploring the balanced binary tree (BBT) schemes for computing the ZRP: firstly an elegant formulae for computation is established; secondly the recursive and iterative algorithms based-on BBT are proposed; thirdly the computational complexity of the algorithms are analyzed rigorously; finally the performance of BBT schemes by testing the running time is verified and validated. Theoretical analysis shows that the computational complexity of balanced binary tree recursive algorithm (BBRTA) and iterative algorithm are exponential and quadratic respectively, which coincides with the running time test very well. Experiments show that the time consumption is about $1\sim 10$ microseconds with different computation platforms for the balanced binary tree iterative algorithm (BBTIA), which is stable and efficient for real-time applications. In the sense of STEM education, the connection of the BBT and ZRP exhibits the beauty and applications of discrete mathematical structure behind the engineering problem, which is worthy of introducing to the college students, computer programmers and optics engineers.

翻译：泽尼克径向多项式（ZRP）在光学设计、成像系统和图像处理系统等应用领域发挥着重要作用。目前，利用计算机程序自动计算ZRP主要有两种数值方案：一种基于定义，其中阶乘运算可能导致溢出问题，且高阶导数计算繁琐；另一种基于递推关系，但此类方法要么不稳定，要么计算复杂度高。本文重点研究基于平衡二叉树（BBT）计算ZRP的方案：首先建立简洁的计算公式；其次提出基于BBT的递归与迭代算法；然后严格分析算法的计算复杂度；最后通过运行时间测试验证BBT方案的性能。理论分析表明，平衡二叉树递归算法（BBRTA）与迭代算法的时间复杂度分别呈指数级和二次方级，这与运行时间测试结果高度吻合。实验显示，在不同计算平台上，平衡二叉树迭代算法（BBTIA）的时间消耗约为1~10微秒，具备稳定高效的特点，适用于实时应用。从STEM教育视角看，BBT与ZRP的结合展示了工程问题背后离散数学结构的美妙与应用，值得向大学生、计算机程序员及光学工程师推广。