I was just given a 5-foot tall Norfolk Island pine. The trunk is bare, and the branches sprout out at the top like a palm. How can I help this?
It sounds like your new Norfolk Island pine (Auracacia heterophylla) has been through a rough time! These plants drop their branches when their cultural requirements - particularly adequate water and light - are not met, and unfortunately the branches will not grow back, and your tree will always somewhat resemble a palm.
Here is some additional information from a previous question about optimal care for these trees:
Norfolk Island pine (Araucaria heterophylla) is a popular indoor conifer, common names are Christmas tree plant, Australian pine, or house pine. In the wild it can mature to 200 feet tall, but rarely exceeds 4 - 6 feet indoors, growing 6 inches a year. New growth--which is normally produced in the spring--is a bright fresh green held until fall when it darkens.
The branches are heavy but they need not be given any extra support, since the main stem turns woody in time. Yes, you can cut back the branches so the plant fits better in the room but no growth is likely to result from the cuts.The droopy nature of this tree is simply a characteristic of this plant. You may need to think carefully about cutting it, since it would remove its natural growth shape.
Grow Araucaria in medium light, but not too far from the window, or needles will fall. A south-or west-facing window, with the plant four feet away from it, would be ideal. Turn the plant every week or so to keep it symmetrical ideally. The temperature for your tree would be best at 45 - 75° F; above 80° F, high humidity will be needed. You can mist spray occasionally.
Water thoroughly keeping soil moist but never let it stand in water. During rest period water moderately, enough to make the potting mixture thoroughly moist, but allow the top inch to dry out between waterings.
Repot about every two or three years, but plants can be moved in spring whenever roots appear on the surface of the mixture or through the bottom of the pot. Five-inch pots are normally the largest necessary, but large specimens may require 6-or even 8-inch pots.
Courtesy of NYBG Plant Information