The bark will generally stay on unless the tree is long dead and has been left outside to weather.
Smooth barked things tend to work better, but since every part of the country/world has different kinds of trees, it's hard to generalize. Aspen here is very plentiful, for example, and apples & crabapples tend to have smooth wood and are fairly common. Here too we have this dreadful invasive tree called the buckthorn, so it's pretty simple to find one and cut it down, but some buckthorns have icky flaky bark - it doesn't bother my kid, but I do need to vacuum up the flakes afterwards. :-)
And trees/branches of the diameter that we use for blocks tend to have thinner & smoother bark then, say, a 12" diameter tree.
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I can not wait till the scroll saw I manifests comes into reality 
