June is a bit quieter than the earlier spring months. The quick spurt of early growth has slowed somewhat, but we still see straggly bits of growth - mostly on deciduous trees - and it should generally be pruned back to two or three leaves. Thinning of the foliage may be necessary to let in more light to the inner branches. Some deciduous trees (but not all!) can be defoliated to reduce leaf size, check the fast elongation of branches, and encourage finer ramification. The second growth will also generally show enhanced fall colors too. Azaleas and quinces can be pruned and pinched now, as they'll have all summer to develop new flower buds.
Conifer and all needle trees also need to be pruned to keep them compact. Pines (especially back and pitch) can have their candles shortened or removed - but leave at least 3-4 needles of the new growth. Pines that feature 2 annual flushes of growth can be fully or partially decandled, but not all pines and not every year - be careful! Junipers can be "finger pruned" to keep them in bounds.
The sun is at its zenith and the days are longest this month. Watch, and perhaps limit the amount of direct sunshine your deciduous trees are receiving, especially Japanese maples - brown-tips season is coming. Provide midday shade and plenty of water to avoid this problem.
Tropicals will enjoy their stay outside. But introduce them to the great outdoors gradually - no sunburn foliage! Watch for spurts of new growth and prune accordingly. Serrissas, among others, don't like direct midday sun, but i you keep them happy with light shade and ample water, they'll reward you with strong, dense growth. Most tropicals are re-potted in summer, as this is their fastest growing season and they will recover quickly.
Fertilize your trees as you did in May. Use appropriate fungicides and pesticides judiciously. Keep all pots free of weeds - not only are they unsightly, they provide cover to insects and they compete with your tree for food.
Careful wiring is possible, but can be difficult on trees that are all leafed out. Wrap a little looser, as trunks and branches are expanding at this time. It's prudent to protect tender bark (azaleas especially) with a layer of raffia under the wire. As usual, check wires often.