Consider a tree $(T, <_T)$ of height $\omega_1$, with countable levels. One can view $T$ as a forcing poset by calling a condition $s\in T$ stronger than $t\in T$ if $t <_T s$.

My question is: when is $T$ proper, as a forcing poset?

If this is too vague, consider the following. The ccc trees are exactly the Suslin trees. However the other common property used to prove properness is countable closure, and no $\omega_1$-tree can be countably closed (a countably closed tree of infinite height has a level of size continuum). Hence we may ask;

Is there a proper $\omega_1$-tree which is not Suslin?

Consider a tree $(T, <_T)$ of height $\omega_1$, with countable levels. One can view $T$ as a forcing poset by calling a condition $s\in T$ stronger than $t\in T$ if $t <_T s$.

My question is: when is $T$ proper, as a forcing poset?

If this is too vague, consider the following. The ccc trees are exactly the Suslin trees. However the other common property used to prove properness is countable closure, and no $\omega_1$-tree can be countably closed (a countably closed tree of infinite height has a level of size continuum). Hence we may ask;

Is there a proper $\omega_1$-tree which is not Suslin?

EDIT

Given Joel's (excellent) answer, I feel I should mention that I'm mainly interested in examples of trees which are branchless, and normal.