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I am just trying to learn how to express common things using mathematical symbols / equations. I would really love to get an intuitive grasp of how to take thoughts about the world and convert them into mathematical expressions. But I don't know how yet and so am wondering how you approach doing this, and/or what this would look like.

When I briefly think of a tree (like an oak tree), I think it has leaves, branches and roots. So what are the ways you would write that if you wanted to start formalizing it into a mathematical expression?

At first thought, it doesn't seem that "leaves are elements of a tree", like elements of a set, and there is more too the tree than just branches and leaves. So not sure how to begin writing that using mathematical symbols. Any ideas?

PVAL-inactive
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Lance
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    Mathematical notation is often for abstract objects. You don't often see a set if you're walking down the street. You'd have to define the tree in terms of well defined mathematical concepts. A tree is not a set. It is a tree. – Jam Aug 12 '14 at 22:14
  • You may take a look here – Hakim Aug 12 '14 at 22:16
  • @Hakim The asker isn't looking at how to model trees, they're asking how to state "a tree is made of leaves..." in mathematical notation. The tag should probably be removed. – Jam Aug 12 '14 at 22:17
  • @EulCan He tagged his question by [tag:mathematical-modeling] and I interpreted his statement: "I would really love to get an intuitive grasp of how to take thoughts about the world and convert them into mathematical expressions." as searching for a mathematical model of trees, anyway. – Hakim Aug 12 '14 at 22:18
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    At the level you're looking at, UML may be a more efficient modeling language than math, and just as abstract. – John Aug 12 '14 at 22:19
  • This has nothing to do with trees, mathematical modeling or abstract algebra... – PVAL-inactive Aug 12 '14 at 22:20
  • An example other than "trees" would probably be more sensible since trees are already an object in mathematics. @Hakim I see what you mean and think that the mathematical-modelling tag suggests that but the question's title seems to be asking how to express statements. – Jam Aug 12 '14 at 22:22
  • You wouldn't. That's not what mathematical symbols are for. – Jack M Aug 12 '14 at 22:36
  • A more "model-able" example is my earlier question (http://math.stackexchange.com/questions/895417/how-to-model-a-real-world-graphical-structure-into-a-mathematical-formulation). Doesn't seem like people want to try to answer that other question, so thought this tree question might be simpler to model or something. I am new to math and am just trying to get a grasp of how you guys think about stuff. Would you mind taking a look at the other question? – Lance Aug 12 '14 at 23:03
  • @LancePollard If you want to know how mathematicians think about problems, the answer is... absolutely not in the way you're apparantly trying to. If you can't tell at a glance how something can be represented mathematically or with set theory, that's probably because it's not worth trying to, and most importantly, representing a problem using symbols won't magically help you solve it. – Jack M Aug 13 '14 at 00:30

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I don't know if I understood correctly your question intent, but first of all, there isn't a unique way of doing this, probably many equivalent definitions can be formulated.

You may define an undirected graph $G=(V,E)$ with a set of vertices and a set of unordered pairs of vertices (the edges). Then, you may define a tree recursively: A tree is an undirected connected graph with a special vertice called $root$ and the root's adjacent subgraphs are also trees.

Then you may define the leaves as the tree vertices (often called nodes) of degree 1, except maybe the root. Branches are the graph's edges. Graphs and Trees are closely related, and Graphs rely heavly on sets and relations formalism.

But you asked about mathematical models, and while real world trees are conveniently modelled with mathematical trees many times, the model is highly dependent of your goals.

As a cartographer engineer, you may model trees in a terrain as cones or spheres...

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