Is a palm tree a tropical tree?

No.

A palm is a monocot (family monocotyledoneae), which is related to corn, grass, bamboo, agave, irises, yucca, etc.

They do not develop rings year to year, and their stems are dramatically different from tree trunks. They do not add bulk to their trunks year to year at all, and their water transportation system is clusters of phloem and zylem packed in a pithy fibrous sheath.

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Palms are neither a softwood (Gymnosperm) nor a hardwood (AngiOSperm).

Palms are members of the Arcaceae family of flowering plants. They have been termed "the princes of the vegetable kingdom."

Of course, due to their enormous size, palms are commonly referred to as 'trees.' If one falls on your car, you're not going to tell the insurance agent that the damage was caused by a flowering plant.

So, while you and I might call them trees, Palms are not scientifically classified as such.
This is similar to asking: Is that animal really a 'dog' or a 'chien' or a 'perro'. It all depends on whether you speak English or French or Spanish. The animal is not really anything all by itself.

Likewise, a palm 'tree' is not really anything, but only what people say it is. The billions of people on the earth today and all who lived before them, have created a Word 'tree' which includes palm trees. This is the classic dictionary definition. Although the Word is different in each language, the concept, the idea of what a 'tree' is, remains the same.

Perennial woody plants having a main trunk, thick bark and distinct crown are trees. In this definition of trees the palms do not have woody trunk, therefore, these are not trees in strict sense.

In very recent times, a small number of people have been devoting themselves to the study of plants. To help them organize their field, they borrow some of our Words and give them new meanings. Within the field of botany, our Word 'tree' is now a technical term which does not include palms. Instead, palms --that don't have wood-- are categorized into the "manicot" class of flowering plants (along with orchids, tulips, onions, sugarcane, bamboo, wheat, and crabgrass).

- So, by botanical definition, a palm tree is not a tree.

- For the billions of non-botanists on the planet, and most of their dictionaries, a palm tree is a tree.

If two different groups use two different definitions, then there are two correct answers. But neither is what it 'really' is because it isn't 'really' anything!

Information which can be found in some relevant books:

According to An encyclopedia of cultivated palms / Robert Lee Riffle and Paul Craft (Portland OR / Cambridge: Timber Press, 2003. ISBN 0881925586) "palms are among the few monocot families with woody tissues". Palms "may be shrublike, treelike or vinelike". "As monocots, palms do not increase their trunk size by growing new wood. Rather, if the trunk enlarges, it does so by expanding the tissues first formed".

Palms are referred to as trees in the following:

Oxford dictionary of plant sciences/ edited by Michael Allaby (Oxford): Oxford University Press, 2004. ISBN 0198608764: page 319)

Plant / Editor-in-chief Janet Marinelli (London: Dorling Kindersley, 2004. ISBN 0751347973: page 356)

The new Royal Horticultural Society dictionary of gardening / Editor-in-chief Anthony Huxley (London: Macmillan, 1992. ISBN 0333474945: Vol.3, page 443)