Uncertainty propagation and categorical data

Suppose that you have a model in which you want to perform uncertainty propagation. For example, consider a model of temperature in an area of the world. To simplify, in this model, Temperature will depend only on the presence of trees and of altitude. Some of the inputs are continuous as altitude and some are discrete as presence of trees or not and what kind of tree. As the presence of tree is a categorical data (presence of maple, palm, pine ...), if i dont know what kind of tree is positionned at an area, i can assume that my tree distribution is a multinomial. By performing some observations, i can assume that P(pine)=0.2,P(maple)=0.1,P(maple)= 0.4 and P(no tree)=0.3. If i perform propagation of these values in a simulation tool, i will get a multinomial as response of my temperature. Usually, uncertainty propagation are based on gaussian propagations and the results are easy for exploitation because gaussians are characterized by mean and variances. For multinomial, i will have the same numbers of means and variances than i have trees catergoricals. My question is the following: what kind of uncertainty models are usually postulated for categorical uncertainty propagations in numerical simulations? How to exploit this kind of results since i can produce by this way a mean and variance for each interval of temperature I get in my model response?